- #1

Mina Farag

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- Homework Statement
- Prove that the following system:

F(x,y,z)=2*x^2+y^2+1−z^2=0

G(x,y,z)=2*x^2+2*y^2−z^2=0

can be solved for z and y as functions of x.

Furthermore, provide the values of the points that allow the parameterization.

- Relevant Equations
- Jacobian of a system, implicit function theorem, Cramer's rule.

**My attempt:**

According to the implicit function theorem as long as the determinant of the jacobian given by ∂(F,G)/∂(y,z) is not equal to 0, the parametrization is possible.

∂(F,G)/∂(y,z)=4yzMeaning that all points where z and y are not equal to 0 are possible parametrizations.

**My friend's solution:**

Same as previous, conclude that according to the implicit function theorem it is all points where z and y are not 0. However, test the possibilities where y=0and z=0

- y=0: by subtracting F−Gyou get 1=0, which is impossible.
- z=0: gives G=2*x^2+2*y^2=0 which gives x=y=0, which is impossible for F, as it gives 1=0.

**My question:**

Is it necessary to consider the possibilities of y=0 and z=0. These possibilities should be impossible as either one makes the determinant of the jacobian equal 0 which if you were to use Cramer's rule to find the partial derivative: ∂(y)/∂(x) would lead to dividing by zero.

Furthermore, it leads to the same conclusion. Is there a reason to why it is important and are there cases where it would not lead to the same conclusion?

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